Final vibrational state distributions

Figure 10. Final vibrational state distributions of CO following the decay of the (0. u2,0) resonances with u2 = 8,..., 11. The arrows mark the highest accessible state for the respective resonance. (Reprinted, with permission of the American Institute of Physics, from Ref. 32.)...

Fig. 6.9. Left-hand side Vibrational excitation function N(ro) and weighting function W(ro) versus the initial oscillator coordinate ro for three values of the coupling parameter e. The equilibrium separation of the free BC molecule is f = 0.403 A and the equilibrium value within the parent molecule is re = 0.481 A. Right-hand side Final vibrational state distributions P(n) for fixed energy E the quantum mechanical and the classical distributions are normalized to the same height at the maxima. The classical distributions are obtained with the help of (6.32). The lowest part of the figure contains also the pure Franck-Condon (FC) distribution (

The final vibrational state distribution essentially reflects the initial distribution of the vibrational coordinate mediated by the excitation function N(ro). 

The final vibrational state distribution of the photofragment manifests the change of its bond length along the dissociation path [see Simons and Yarwood (1963) and Mitchell and Simons (1967) for early references]. Let us consider the dissociation of a linear triatomic molecule ABC described by Jacobi coordinates R and r as defined in Figure 2.1. In classical mechanics, it is the force... 

The main purpose of this chapter is to emphasize the intimate relation between the topology of the dissociative PES and the vibrational excitation of the fragment molecule. In Sections 9.1 and 9.2 we consider exclusively direct processes. The photodissociation of symmetric molecules with two equivalent product channels is the topic of Section 9.3. Finally, vibrational state distributions following the decay of a long-lived intermediate complex will be discussed in Section 9.4. The theory... 

Fig. 9.3. Energy dependence of final vibrational state distributions for the cases (a) and (b) illustrated in Figure 9.1. The total energy is normalized such that E = 0 corresponds to A + BC(re).

Fig. 9.4. Left-hand side Representation of an inelastic potential energy surface of the form (6.35) with e = 0.2. The heavy arrows represent two characteristic trajectories starting at the respective FC points. Right-hand side The corresponding final vibrational state distributions.

The vibrational reflection principle outlined in Section 6.4 provides a simple, but yet quantitative explanation of final vibrational state distributions and their variation with the coupling strength and the total energy. The central quantity is the vibrational excitation function N(ro). It comprehensively manifests the dynamical details of the fragmentation process in the upper electronic state. Usually, one needs only very few trajectories to construct N(ro) which makes the simple classical theory outlined in Section 6.4 very efficient for calculating and understanding final state distributions. This is particularly beneficial for fitting experimental data. 

Fig. 9.7. (a) Final vibrational state distribution in the umbrella mode of CH3... 

The photodissociation of symmetric molecules such as H2O, H2S, and O3 illustrates particularly clearly the close relation between the change of the molecular configuration along the dissociation path and the final vibrational state distribution. Figure 9.9 shows the two-dimensional PES... 

As a consequence of the strong interaction of the two electronic states the shape of the dissociative PES in the saddle point region differs noticeably from the dissociative PES for H20(A) [see Figure 15.6(a)], The saddle is considerably narrower than for H2O. Although the differences are rather subtle, they explain qualitatively the dissimilarity of the final vibrational state distributions. Trajectories starting near the tran-... 

The right-hand part of Figure 9.12 depicts the final vibrational state distributions of NO obtained in a three-dimensional wavepacket calculation. They reveal a weak propensity for a final vibrational state n = n — 2. Excitation of vibrational state n in the HONO (Si) complex leads preferentially to NO products in vibrational state (n — 2). Because the intermediate potential well is rather deep, 0.362 eV, the breakup of... 

Figure 6.13 showed an example of applying Eq. (6.32) to the final vibrational state distribution in the F + H2 reaction. The observed distribution is quahtatively different from the prior, as is only to be expected for a direct reaction. Yet the surprisal has a simple (linear) dependence on the fraction of the available energy that is in the vibration. The simplicity of a linear surprisal is not caused by the semilogarithmic nature of the plot. Figure 1.3 showed the actual rotational state populations of HD in several vibrational manifolds from the H + D2 reaction. The fit to a linear surprisal is close. 

L. Collisions of the second kind. The one-dimensional potentials that you sketched in Problem K overlook the role of the vibration of 02- Examine this point in at least one of two ways, (a) Efraw potential energy surfaces that are functions of both the Na—O2 distance R and the 0—0 distance r. (b) Draw onedimensional potential energy curves where for each curve both the electronic state of Na and the vibrational state of O2 are fixed (such potentials are diabatic in two senses). Discuss the dynamics of the quenching and show why we expect that O2 will be vibrationally excited after a quenching collision. The final vibrational state distribution in processes in which an electronic state change occurs are often quite close to the prior limit. Can your considerations above rationalize why this will be so ... 

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